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Overview

More than three centuries after its creation, calculus remains a dazzling intellectual achievement and the gateway into higher mathematics. This book charts its growth and development by sampling from the work of some of its foremost practitioners, beginning with Isaac Newton and Gottfried Wilhelm Leibniz in the late seventeenth century and continuing to Henri Lebesgue at the dawn of the twentieth--mathematicians whose achievements are comparable to those of Bach in music or Shakespeare in literature. William Dunham lucidly presents the definitions, theorems, and proofs. "Students of literature read Shakespeare; students of music listen to Bach," he writes. But this tradition of studying the major works of the "masters" is, if not wholly absent, certainly uncommon in mathematics. This book seeks to redress that situation.

Like a great museum, The Calculus Gallery is filled with masterpieces, among which are Bernoulli's early attack upon the harmonic series (1689), Euler's brilliant approximation of pi (1779), Cauchy's classic proof of the fundamental theorem of calculus (1823), Weierstrass's mind-boggling counterexample (1872), and Baire's original "category theorem" (1899). Collectively, these selections document the evolution of calculus from a powerful but logically chaotic subject into one whose foundations are thorough, rigorous, and unflinching--a story of genius triumphing over some of the toughest, most subtle problems imaginable.

Anyone who has studied and enjoyed calculus will discover in these pages the sheer excitement each mathematician must have felt when pushing into the unknown. In touring The Calculus Gallery, we can see how it all came to be.

What People Are Saying

Henry Pollak

The Calculus Gallery is one of the best efforts at mathematical exposition I have ever read! Dunham presents in detail and in his own words the sequence of ideas of classical giants of mathematics, but each new idea is described in modern terms and notation, so I had absolutely no trouble following along. Furthermore—and this is an astounding achievement—the entire work has a tightly woven development. If it were a detective story I would say it had a plot with no loose ends. An amazing feat. I wish I could plan a single lecture, never mind a course or a book, that well!
— Henry Pollak, Teachers College, Columbia University

Thomas Banchoff

What a fine resource! All of the famous functions that have shaped calculus and analysis parade before the reader in the original words of their creators. Bill Dunham has produced an excellent volume that teachers and students will enjoy and appreciate.
— Thomas Banchoff, Brown University

Robin Wilson

Bill Dunham has done it again. The Calculus Gallery is a masterly journey through the works of thirteen mathematicians who formulated, formalised, and reformed the calculus into the modern analysis we learn today. Readers of his earlier books have learned to expect a clarity of exposition that few others can attain: they will not be disappointed.
— Robin Wilson, author of "Four Colors Suffice"

Barry Cipra

This is an excellent book—an amazing mathematical page-turner. William Dunham has done the seemingly impossible: he has taken some difficult, advanced mathematics and, without sacrificing the technical details, written a lively, readable book about it.
— Barry Cipra, author of "Misteaks . . . and How to Find Them Before the Teacher Does"

Annette Imhausen

Pedagogically excellent and extremely well written, The Calculus Gallery bridges the gap between general histories and detailed studies of individual mathematicians. Dunham has described mathematical developments in an engaging style rarely found in literature of this kind.
— Annette Imhausen, Trinity Hall, Cambridge

Franz Lemmermeyer

A welcome addition to the literature. The idea of presenting a 'museum of mathematics' is new. It allows the author to present a nonstandard selection of theorems, so that even mathematicians with a strong historical background will learn a few things.
— Franz Lemmermeyer, Bilkent University, author of "Reciprocity Laws: From Euler to Eisenstein"

Barry Cipra

This is an excellent book--an amazing mathematical page-turner. William Dunham has done the seemingly impossible: he has taken some difficult, advanced mathematics and, without sacrificing the technical details, written a lively, readable book about it.
— Barry Cipra, author of "Misteaks . . . and How to Find Them Before the Teacher Does"

Henry Pollak

The Calculus Gallery is one of the best efforts at mathematical exposition I have ever read! Dunham presents in detail and in his own words the sequence of ideas of classical giants of mathematics, but each new idea is described in modern terms and notation, so I had absolutely no trouble following along. Furthermore--and this is an astounding achievement--the entire work has a tightly woven development. If it were a detective story I would say it had a plot with no loose ends. An amazing feat. I wish I could plan a single lecture, never mind a course or a book, that well!
— Henry Pollak, Teachers College, Columbia University

Editorial Reviews

Zentralblatt MATH Database

Zentralblatt MATH

Science
- Judith V. Grabiner

The Calculus Gallery is a wonderful book. The style is inviting; the explanations are clear and accessible. . . . Mathematicians, scientists, and historians alike can learn much that is interesting, much that is mathematically significant, and a good deal that is both.

American Scientist
- Victor J. Katz

[A] brilliant book. . . . I predict that Dunham's book will itself come to be considered a masterpiece in its field.

Mathematics Teacher
- Diane M. Spresser

A joy to read, The Calculus Gallery showcases one of the great intellectual pursuits of all time and, in the words of John von Neumann, 'the first achievement of modern mathematics.' Thirteen scholars, beginning with Newton and Leibniz, who gave birth to calculus in the seventeenth century, are featured in this sequential development of the important ideas that shaped calculus as we know it and gave rise to modern analysis. . . . [I]t is a lovely and engaging gallery of the 'masters' that belongs in the library of everyone who seriously teaches or studies the subject.

Zentralblatt MATH
- Eberhard Knobloch

A fascinating, competent visit too the calculus gallery.

From the Publisher

One of Choice's Outstanding Academic Titles for 2005

"The Calculus Gallery is a wonderful book. The style is inviting; the explanations are clear and accessible. . . . Mathematicians, scientists, and historians alike can learn much that is interesting, much that is mathematically significant, and a good deal that is both."--Judith V. Grabiner, Science

"[A] brilliant book. . . . I predict that Dunham's book will itself come to be considered a masterpiece in its field."--Victor J. Katz, American Scientist

"What distinguishes this selection is it truly provides a history of mathematics, not just a history of mathematicians. . . . If a better historical treatment of the development of the calculus is available, this reviewer has yet to see it. . . . Essential."--Choice

"A joy to read, The Calculus Gallery showcases one of the great intellectual pursuits of all time and, in the words of John von Neumann, 'the first achievement of modern mathematics.' Thirteen scholars, beginning with Newton and Leibniz, who gave birth to calculus in the seventeenth century, are featured in this sequential development of the important ideas that shaped calculus as we know it and gave rise to modern analysis. . . . [I]t is a lovely and engaging gallery of the 'masters' that belongs in the library of everyone who seriously teaches or studies the subject."--Diane M. Spresser, Mathematics Teacher

Science

The Calculus Gallery is a wonderful book. The style is inviting; the explanations are clear and accessible. . . . Mathematicians, scientists, and historians alike can learn much that is interesting, much that is mathematically significant, and a good deal that is both.
— Judith V. Grabiner

American Scientist

[A] brilliant book. . . . I predict that Dunham's book will itself come to be considered a masterpiece in its field.
— Victor J. Katz

Choice

What distinguishes this selection is it truly provides a history of mathematics, not just a history of mathematicians. . . . If a better historical treatment of the development of the calculus is available, this reviewer has yet to see it. . . . Essential.

Mathematics Teacher

A joy to read, The Calculus Gallery showcases one of the great intellectual pursuits of all time and, in the words of John von Neumann, 'the first achievement of modern mathematics.' Thirteen scholars, beginning with Newton and Leibniz, who gave birth to calculus in the seventeenth century, are featured in this sequential development of the important ideas that shaped calculus as we know it and gave rise to modern analysis. . . . [I]t is a lovely and engaging gallery of the 'masters' that belongs in the library of everyone who seriously teaches or studies the subject.
— Diane M. Spresser

Science

The Calculus Gallery is a wonderful book. The style is inviting; the explanations are clear and accessible. . . . Mathematicians, scientists, and historians alike can learn much that is interesting, much that is mathematically significant, and a good deal that is both.
— Judith V. Grabiner

Mathematics Teacher

A joy to read, The Calculus Gallery showcases one of the great intellectual pursuits of all time and, in the words of John von Neumann, 'the first achievement of modern mathematics.' Thirteen scholars, beginning with Newton and Leibniz, who gave birth to calculus in the seventeenth century, are featured in this sequential development of the important ideas that shaped calculus as we know it and gave rise to modern analysis. . . . [I]t is a lovely and engaging gallery of the 'masters' that belongs in the library of everyone who seriously teaches or studies the subject.
— Diane M. Spresser

Related Subjects

Meet the Author

William Dunham, Koehler Professor of Mathematics at Muhlenberg College, is the author of "Journey Through Genius: The Great Theorems of Mathematics"; "The Mathematical Universe"; and "Euler: The Master of Us All". He has received the Mathematical Association of America's George Polya, Trevor Evans, and Lester R. Ford awards, as well as its Beckenbach Prize for expository writing.

The Calculus Gallery

Masterpieces from Newton to Lebesgue

PRINCETON UNIVERSITY PRESS

Isaac Newton (1642–1727) stands as a seminal figure not just in mathematics but in all of Western intellectual history. He was born into a world where science had yet to establish a clear supremacy over medieval superstition. By the time of his death, the Age of Reason was in full bloom. This remarkable transition was due in no small part to his own contributions.

For mathematicians, Isaac Newton is revered as the creator of calculus, or, to use his name for it, of "fluxions." Its origin dates to the mid-1660s when he was a young scholar at Trinity College, Cambridge. There he had absorbed the work of such predecessors as René Descartes (1596–1650), John Wallis (1616–1703), and Trinity's own Isaac Barrow (1630–1677), but he soon found himself moving into uncharted territory. During the next few years, a period his biographer Richard Westfall characterized as one of "incandescent activity," Newton changed forever the mathematical landscape. By 1669, Barrow himself was describing his colleague as "a fellow of our College and very young ... but of an extraordinary genius and proficiency".

In this chapter, we look at a few of Newton's early achievements: his generalized binomial expansion for turning certain expressions into infinite series, his technique for finding inverses of such series, and his quadrature rule for determining areas under curves. We conclude with a spectacular consequence of these: the series expansion for the sine of an angle. Newton's account of the binomial expansion appears in his epístola prior, a letter he sent to Leibniz in the summer of 1676 long after he had done the original work. The other discussions come from Newton's 1669 treatise De analysi per aequatíones numero terminorum infinitas, usually called simply the De analysi.

Although this chapter is restricted to Newton's early work, we note that "early" Newton tends to surpass the mature work of just about anyone else.

Generalized Binomial Expansion

By 1665, Isaac Newton had found a simple way to expand—his word was "reduce"—binomial expressions into series. For him, such reductions would be a means of recasting binomials in alternate form as well as an entryway into the method of fluxions. This theorem was the starting point for much of Newton's mathematical innovation.

As described in the epistola prior, the issue at hand was to reduce the binomial (P + PQ)m/n and to do so whether m/n "is integral or (so to speak) fractional, whether positive or negative". This in itself was a bold idea for a time when exponents were sufficiently unfamiliar that they had first to be explained, as Newton did by stressing that "instead of √a, [cube root of (a)], [cube root of (a5)], etc. I write a1/2, a1/3, a5/3, and instead of 1/a, 1/aa, 1/a3, I write a-1, a-2, a-3". Apparently readers of the day needed a gentle reminder.

Newton discovered a pattern for expanding not only elementary binomials like (1 + x)5 but more sophisticated ones like [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The reduction, as Newton explained to Leibniz, obeyed the rule

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where each of A, B, C, ... represents the previous term, as will be illustrated below. This is his famous binomial expansion, although perhaps in an unfamiliar guise.

To identify A, B, C, and the rest, we recall that each is the immediately preceding term. Thus, A = (c2)1/2 = c, giving us

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Likewise B is the previous term—i.e., B = x2/2c —so at this stage we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The analogous substitutions yield C = x4/8c3 and then D = x6/16c5. Working from left to right in this fashion, Newton arrived at

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Obviously, the technique has a recursive flavor: one finds the coefficient of x8 from the coefficient of x6, which in turn requires the coefficient of x4, and so on. Although the modern reader is probably accustomed to a "direct" statement of the binomial theorem, Newton's recursion has an undeniable appeal, for it streamlines the arithmetic when calculating a numerical coefficient from its predecessor.

For the record, it is a simple matter to replace A, B, C, ... by their equivalent expressions in terms of P and Q, then factor the common pm/n from both sides of (1), and so arrive at the result found in today's texts:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

Newton likened such reductions to the conversion of square roots into infinite decimals, and he was not shy in touting the benefits of the operation. "It is a convenience attending infinite series," he wrote in 1671,

that all kinds of complicated terms ... may be reduced to the class of simple quantities, i.e., to an infinite series of fractions whose numerators and denominators are simple terms, which will thus be freed from those difficulties that in their original form seem'd almost insuperable.

To be sure, freeing mathematics from insuperable difficulties is a worthy undertaking.

One additional example may be helpful. Consider the expansion of 1/√ 1 - x2, which Newton put to good use in a result we shall discuss later in the chapter. We first write this as (1 - x2)-1/2, identify m = - 1, n = 2, and Q = -x2, and apply (2):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

Newton would "check" an expansion like (3) by squaring the series and examining the answer. If we do the same, restricting our attention to terms of degree no higher than x8, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where all of the coefficients miraculously turn out to be 1 (try it!). The resulting product, of course, is an infinite geometric series with common ratio x2 which, by the well-known formula, sums to 1/1 - x2. But if the square of the series in (3) is 1/1 - x2, we conclude that that series itself must be 1/&#8730 1 - x2. Voila!

Newton regarded such calculations as compelling evidence for his general result. He asserted that the "common analysis performed by means of equations of a finite number of terms" may be extended to such infinite expressions "albeit we mortals whose reasoning powers are confined within narrow limits, can neither express nor so conceive all the terms of these equations, as to know exactly from thence the quantities we want".

Inverting Series

Having described a method for reducing certain binomials to infinite series of the form z = A + Bx + Cx2 + Dx3 + ···, Newton next sought a way of finding the series for x in terms of z. In modern terminology, he was seeking the inverse relationship. The resulting technique involves a bit of heavy algebraic lifting, but it warrants our attention for it too will appear later on. As Newton did, we describe the inversion procedure by means of a specific example.

Beginning with the series z = x - x2 + x3 - x4 + ···, we rewrite it as

(x - x2 + x3 - x4 + ···) - z = 0 (4)

and discard all powers of x greater than or equal to the quadratic. This, of course, leaves x - z = 0, and so the inverted series begins as x = z.

Newton was aware that discarding all those higher degree terms rendered the solution inexact. The exact answer would have the form x = z + p, where p is a series yet to be determined. Substituting z + p for x in (4) gives

[(z + p) - (z + p)2 + (z + p)3 - (z + p) + ···]- z = 0,

which we then expand and rearrange to get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

Next, jettison the quadratic, cubic, and higher degree terms in p and solve to get

p = z2 - z3 + z4 - z5 + ···/ 1 - 2z + 3z2 - 4z3 + ···.

Newton now did a second round of weeding, as he tossed out all but the lowest power of z in numerator and denominator. Hence p is approximately z2/1, so the inverted series at this stage looks like x = z + p = z + z2.

But p is not exactly z2. Rather, we say p = z2 + q, where q is a series to be determined. To do so, we substitute into (5) to get

The process would be continued by substituting q = z3 + r into (6). Newton, who had a remarkable tolerance for algebraic monotony, seemed able to continue such calculations ad infinitum (almost). But eventually even he was ready to step back, examine the output, and seek a pattern. Newton put it this way: "Let it be observed here, by the bye, that when 5 or 6 terms ... are known, they may be continued at pleasure for most part, by observing the analogy of the progression".

In what sense can this be trusted? After all, Newton discarded most of his terms most of the time, so what confidence remains that the answer is correct?

Again, we take comfort in the following "check." The original series z = x - x2 + x3 - x4 + ··· is geometric with common ratio - x, and so in closed form z = x/1 + x. Consequently, x = z/1 - z, which we recognize to be the sum of the geometric series z + z2 + z3 + z4 + z5 + ···. This is precisely the result to which Newton's procedure had led us. Everything seems to be in working order.

The techniques encountered thus far—the generalized binomial expansion and the inversion of series—would be powerful tools in Newton's hands. There remains one last prerequisite, however, before we can truly appreciate the master at work.

Quadrature Rules From The De Analysi

In his De analysi of 1669, Newton promised to describe the method "which I had devised some considerable time ago, for measuring the quantity of curves, by means of series, infinite in the number of terms". This was not Newton's first account of his fluxional discoveries, for he had drafted an October 1666 tract along these same lines. The De analysi was a revision that displayed the polish of a maturing thinker. Modern scholars find it strange that the secretive Newton withheld this manuscript from all but a few lucky colleagues, and it did not appear in print until 1711, long after many of its results had been published by others. Nonetheless, the early date and illustrious authorship justify its description as "perhaps the most celebrated of all Newton's mathematical writings".

The treatise began with a statement of the three rules for "the quadrature of simple curves." In the seventeenth century, quadrature meant determination of area, so these are just integration rules.

Rule 1. The quadrature of simple curves: If y = axm/n is the curve AD, where a is a constant and m and n are positive integers, then the area of region ABD is an/m + n x(m+n)/n (see figure 1.1).

A modern version of this would identify A as the origin, B as (x, 0), and the curve as y = atm/n. Newton's statement then becomes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which is just a special case of the power rule from integral calculus.

Only at the end of the De analysi did Newton observe, almost as an afterthought, that "an attentive reader" would want to see a proof for Rule 1. Attentive as always, we present his argument below.

Again, let the curve be AD with AB = x and BD = y, as shown in figure 1.2. Newton assumed that the area ABD beneath the curve was given by an expression z written in terms of x. The goal was to find a corresponding formula for y in terms of x. From a modem vantage point, he was beginning with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and seeking y = y(x). His derivation blended geometry, algebra, and fluxions before ending with a few dramatic flourishes.

At the outset, Newton let β be a point on the horizontal axis a tiny distance o from B. Thus, segment Aβ has length x + o. He let z be the area ABD, although to emphasize the functional relationship we shall take the liberty of writing z = z(x). Hence, z(x + o) is the area Aβδ under the curve. Next he introduced rectangle BβHK of height v = BK = βH, the area of which he stipulated to be exactly that of region BβδD beneath the curve. In other words, the area of BβδD was to be ov.

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